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Theorem peano2nn 7707
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
peano2nn (A ℕ → (A + 1) ℕ)

Proof of Theorem peano2nn
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfnn2 7697 . . . . . 6 ℕ = {x ∣ (1 x y x (y + 1) x)}
21eleq2i 2101 . . . . 5 (A ℕ ↔ A {x ∣ (1 x y x (y + 1) x)})
3 elintg 3614 . . . . 5 (A ℕ → (A {x ∣ (1 x y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)}A z))
42, 3syl5bb 181 . . . 4 (A ℕ → (A ℕ ↔ z {x ∣ (1 x y x (y + 1) x)}A z))
54ibi 165 . . 3 (A ℕ → z {x ∣ (1 x y x (y + 1) x)}A z)
6 vex 2554 . . . . . . . 8 z V
7 eleq2 2098 . . . . . . . . 9 (x = z → (1 x ↔ 1 z))
8 eleq2 2098 . . . . . . . . . 10 (x = z → ((y + 1) x ↔ (y + 1) z))
98raleqbi1dv 2507 . . . . . . . . 9 (x = z → (y x (y + 1) xy z (y + 1) z))
107, 9anbi12d 442 . . . . . . . 8 (x = z → ((1 x y x (y + 1) x) ↔ (1 z y z (y + 1) z)))
116, 10elab 2681 . . . . . . 7 (z {x ∣ (1 x y x (y + 1) x)} ↔ (1 z y z (y + 1) z))
1211simprbi 260 . . . . . 6 (z {x ∣ (1 x y x (y + 1) x)} → y z (y + 1) z)
13 oveq1 5462 . . . . . . . 8 (y = A → (y + 1) = (A + 1))
1413eleq1d 2103 . . . . . . 7 (y = A → ((y + 1) z ↔ (A + 1) z))
1514rspcva 2648 . . . . . 6 ((A z y z (y + 1) z) → (A + 1) z)
1612, 15sylan2 270 . . . . 5 ((A z z {x ∣ (1 x y x (y + 1) x)}) → (A + 1) z)
1716expcom 109 . . . 4 (z {x ∣ (1 x y x (y + 1) x)} → (A z → (A + 1) z))
1817ralimia 2376 . . 3 (z {x ∣ (1 x y x (y + 1) x)}A zz {x ∣ (1 x y x (y + 1) x)} (A + 1) z)
195, 18syl 14 . 2 (A ℕ → z {x ∣ (1 x y x (y + 1) x)} (A + 1) z)
20 nnre 7702 . . . 4 (A ℕ → A ℝ)
21 1red 6840 . . . 4 (A ℕ → 1 ℝ)
2220, 21readdcld 6852 . . 3 (A ℕ → (A + 1) ℝ)
231eleq2i 2101 . . . 4 ((A + 1) ℕ ↔ (A + 1) {x ∣ (1 x y x (y + 1) x)})
24 elintg 3614 . . . 4 ((A + 1) ℝ → ((A + 1) {x ∣ (1 x y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)} (A + 1) z))
2523, 24syl5bb 181 . . 3 ((A + 1) ℝ → ((A + 1) ℕ ↔ z {x ∣ (1 x y x (y + 1) x)} (A + 1) z))
2622, 25syl 14 . 2 (A ℕ → ((A + 1) ℕ ↔ z {x ∣ (1 x y x (y + 1) x)} (A + 1) z))
2719, 26mpbird 156 1 (A ℕ → (A + 1) ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300   cint 3606  (class class class)co 5455  cr 6710  1c1 6712   + caddc 6714  cn 7695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7696
This theorem is referenced by:  peano2nnd  7710  nnind  7711  nnaddcl  7715  2nn  7855  3nn  7856  4nn  7857  5nn  7858  6nn  7859  7nn  7860  8nn  7861  9nn  7862  10nn  7863  nneoor  8116  fzonn0p1p1  8839  expp1  8916
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